Properties

Label 1-24e2-576.133-r0-0-0
Degree $1$
Conductor $576$
Sign $-0.825 + 0.564i$
Analytic cond. $2.67493$
Root an. cond. $2.67493$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.793 + 0.608i)5-s + (−0.258 + 0.965i)7-s + (0.991 − 0.130i)11-s + (−0.130 + 0.991i)13-s i·17-s + (−0.923 + 0.382i)19-s + (0.258 + 0.965i)23-s + (0.258 − 0.965i)25-s + (−0.608 + 0.793i)29-s + (0.5 − 0.866i)31-s + (−0.382 − 0.923i)35-s + (−0.923 − 0.382i)37-s + (0.258 + 0.965i)41-s + (−0.991 + 0.130i)43-s + (−0.866 + 0.5i)47-s + ⋯
L(s)  = 1  + (−0.793 + 0.608i)5-s + (−0.258 + 0.965i)7-s + (0.991 − 0.130i)11-s + (−0.130 + 0.991i)13-s i·17-s + (−0.923 + 0.382i)19-s + (0.258 + 0.965i)23-s + (0.258 − 0.965i)25-s + (−0.608 + 0.793i)29-s + (0.5 − 0.866i)31-s + (−0.382 − 0.923i)35-s + (−0.923 − 0.382i)37-s + (0.258 + 0.965i)41-s + (−0.991 + 0.130i)43-s + (−0.866 + 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.825 + 0.564i$
Analytic conductor: \(2.67493\)
Root analytic conductor: \(2.67493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 576,\ (0:\ ),\ -0.825 + 0.564i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2121081971 + 0.6857404448i\)
\(L(\frac12)\) \(\approx\) \(0.2121081971 + 0.6857404448i\)
\(L(1)\) \(\approx\) \(0.7511285643 + 0.2963960791i\)
\(L(1)\) \(\approx\) \(0.7511285643 + 0.2963960791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.793 + 0.608i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
11 \( 1 + (0.991 - 0.130i)T \)
13 \( 1 + (-0.130 + 0.991i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.923 + 0.382i)T \)
23 \( 1 + (0.258 + 0.965i)T \)
29 \( 1 + (-0.608 + 0.793i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (0.258 + 0.965i)T \)
43 \( 1 + (-0.991 + 0.130i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.382 + 0.923i)T \)
59 \( 1 + (-0.793 + 0.608i)T \)
61 \( 1 + (0.608 - 0.793i)T \)
67 \( 1 + (-0.991 - 0.130i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 + (0.793 + 0.608i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.91171956605024115826496352813, −22.29498619594166502649075751455, −21.019759272449661672127488852978, −20.30496663729117059845365047055, −19.54648199262769241503011432487, −19.11408844658959061733357701510, −17.52891609539879359119363875240, −17.074973173692075850835769816739, −16.275662264178104394090109540454, −15.24492870522037262322920266905, −14.6051422457370218616533702400, −13.35463392123656067806100403176, −12.70638485161448603515216104215, −11.86988632672110811252907490306, −10.79509858189016098264740554264, −10.10353430558634480353541186477, −8.82225420243921330098843879787, −8.174353744166908044995599518796, −7.12750282050454008151853321048, −6.29983435037605121801903581663, −4.90173513799336292202411942013, −4.07708873643580332904573341803, −3.3109514956429265247979550812, −1.610647164720834223296935029929, −0.373531197811871191807233132421, 1.6940418720738943954020293219, 2.89299716678211298402810325077, 3.80651979693962518173527115131, 4.84538203236722268133929541990, 6.17561980112379346426795678528, 6.837432438240827996329842185562, 7.86555905160919550767710245746, 8.97648756445796791714367629748, 9.55608155595098045721317269291, 10.9313247672585304547750680809, 11.72024160114830338842729395306, 12.16460652566559627966597173939, 13.43914699735093505332919438509, 14.51777639250810306763361362402, 15.01726970079729916750509755155, 16.00441431240366075007346238262, 16.69255940638057710319422213868, 17.84748394771146151434467989636, 18.86668491586205123119460840723, 19.14247193432188167298331349405, 20.06962311461794811413982905249, 21.25936049065357369421425924183, 21.97076054914318258873416574593, 22.68748075208053400708446902303, 23.45341014313376052186931252624

Graph of the $Z$-function along the critical line